First, your utility function is strictly increasing in both goods, so you know that your budget will bind so you can set I=pxx+pyx

Second, your utility function is known as "quasilinear", where the good y is linear in your utility and the other good is there in some increasing function not connected to y. That's why the marginal utility for y is just a constant.

Finally, solve for x, where x is a function of prices, then plug that into the budget constraint to solve for y. y should be a function of prices and income.

Since you showed some work:

Set the ratio of MUs to the ratio of prices: $MU_x/MU_y=p_x/p_y$, yielding a demand for $x=\frac{p_y^2}{4p_x^2}$

Now plug that into the budget, yielding the demand for $y= \frac{I}{p_y}-\frac{p_y}{4p_x}$

$\begingroup$so based off of this my demand function for good x would be equivalent to X=Py^2/4Px^2. Then plugging it into the budget constraint of I=PxX+PyY, Y would give Y=-(Py^2-4PxI)/4PyPx. Is this correct @frage_man?$\endgroup$
– T.HAug 21 '16 at 3:00